Albert Alex ZevelevFigures

Orders:A relation from A to B is a subset of the cartesian product R A B.Thus the set of all relations is P(AB).

Inj(A;B)Injections

Bij(A;B)Bijections

BAfunctions

P(AB)relations

Surj(A;B)Surjections

Note that the set Bij(A;B) 6= iff |A| = |B|.Note: surjection f : X Y iff #X #Y injection f : X Y iff #X #Y

Pre-ordered setReflexive, Transitive (R.T.)

POSETReflexive, Anti-symmetric, Transitive (R.A.T.)

LatticePOSET with the LUB and GLB properties

LOSET (Linear Order, Total Order)A POSET with completeness

Well Ordered SetA LOSET where every non-empty set has aleast element

Example: Consider the set of matrices Mn(R) along with the order PDwhere X PD Y if X Y is positive definite. (Mn(R),PD) is a partialorder.This order is used in the Gauss-Markov theorem.

1

2Add examples of Op() and op() convergence from Wooldridge.

We can define an equivalence class on any relation.Every equivalence class in an equivalence relation contains exactly one point[x] = x, because it is anti-symmetric.In general for any anti-symmetric relation (X,%) every equivalence class isa singleton.If (X,%) is complete and symmetric then [x] = X (all elements are indiffer-ent) so %= X X.

Complete orders

Pre-ordered setReflexive, Transitive

Weakly ordered set (Rational)Complete (thus Reflexive), Transitive

LOSET

Well Ordered Set

A relation (X,%) is reflexive it contains the diagonal X %.Every complete relation is reflexive.See diagram summarizing relations in Michael Carter.Any relation (X,%) has a symmetric part and an asymmetric part which are disjoint: %= .When the symmetric part of a relation is empty = , we call the relation(X,%) asymmetric.The symmetric part of a pre-order is by definition an equivalence relation.= (% %1) and = (% \ ).Monotone comparative statics (MCS) extends Topkis Theorem from func-tions defined on the product of losets to more general functions defined onthe products of lattices and posets.Topkis Thm: f : X

Loset T

Loset R

MCS: f : XLattice

(variables)

TPoset

(parameters)

R

From Casella & Berger (2002) p. 78, exercise (2.15), (Betteley, 1977):Let (X Y ) min(X,Y ) , and (X Y ) max(X,Y ).

3Analogous to the probability law: P (A B) = P (A) + P (B) P (A B),we haveE[X Y ] = E[X] + E[Y ] E[X Y ]Hint: establish the fact (X Y ) + (X Y ) = X + YThus the expectation operator is modular!

FFTWE

Feasible Allocations

Weakly Pareto Efficient

Pareto Efficient

Walrasian Equilibrium

4Topological Space (X, )

Hausdorff Space T2

Normal Space T4

Metric Space (X, d)

Normed Space (X, )

Inner Product Space (X,< , >)

Homeomorphism is to (X, ) as isometry is to (X, d).

The following diagram should be nested inside of the diagram above:

Complete Metric Space

Banach Space

Hilbert Space

5Remember every compact metric space is complete.A finite topological space is metrizable iff it has the discrete topology (wiki).Any Hausdorff T2 topology on a finite topological space is discrete.Every discrete metric space is complete.If X is a finite dimensional vector space then all norms induce the same(Euclidean) topology as Rn.Points are weakly closer to the origin when p is bigger:x1 x2 . . . xp x for x Rn.So when p is bigger, the distance from the origin is weakly smaller. All thesenorms are equal is the point lies on one of the axes (CSZ).To a cab driver in Manhattan using d1 is much more practical than d2because he cant travel diagonally (show this).Show how d2 follows from the triangle inequality.

All Sequences XN

Bounded Sequences B(X), `(X)

Cauchy Sequences C(X)

Convergent Sequences Conv(X)

Eventually Constant Sequences

To discuss bounded sequences and Cauchy sequences we need a metric, todiscuss convergent sequences we need a topology.A sequence is bounded iff its image is bounded.(X, d) is a complete Metric Space iff C(X) = Conv(X) i.e. ever Cauchysequence converges.In the indiscrete topology there are very few open sets so its easy for se-quences to converge, in fact every sequence converges (X, )N = Conv(X).In the discrete topology there are too many open sets its difficult for asequence to converge, the only convergent sequences are the eventually con-stant. (Chapter 3)There are Cauchy sequences in Q that get arbitrarily close to

2 without

6every being equal to

2, thus Conv(Q) ( C(Q).In CSZ p. 312 C(N;R) RN, and Cb(N) C(N)

2 notions about the completeness of R:We know that R is complete because every Cauchy sequence of real numbersconverges to a real number. So if a Cauchy sequence which comes arbitrar-ily close to a point, that point will exist in R.The completeness axiom for the real numbers says that any subset S Rthat has a lower bound must have a greatest lower bound in R. (This impliesthe same for upper bounded sets and supremum.) So the set {q Q : q2 > 2}has many lower bounds (for example 0) but it also has a (unique) greatestlower bound in R namely

2. Thus R is a complete ordered field, whereas

Q is not. In fact it is known that R is the only complete ordered field.

Let s be the set of real sequences where all but finitely many terms arezero. The real sequence {xn} `p if

n=1 |xn|p < . Note that if p q

then `p `q. Let c0 be the set of all real sequences converging to zero, andlet c be the set of all convergent real sequences. Also let ` be the set ofall bounded real sequences, so {xn} ` if sup

nN|xn| < . We have the

following for p q (from Border and from Milman):

s `p `q c0 c ` RN

Note that if a real sequence does not converge to zero {xn} / c0 then its sumcannot be finite. However suppose xn c then the sequence {xn c} 0.

7Continuous functions C0

Uniformly Continuous

Absolutely Continuous

Lipschitz Continuous

Bounded Derivative

Also make a note on Holder Continuity. Topological Spaces

We can take any non-empty set X and add algebraic structure, geomet-ric structure, order structure and measurable structure. For example weusually work with (R, , E

geometric

, order

, ,+Algebraic

, B(R) measurable

), we use this structure

so often that we sometimes take it for granted and simply write R.It is convenient that these structures are consistent: the order topology in-duced by the usual order is the euclidean topology, the product topology isthe euclidean topology on Rk, the product sigma-algebra is B(Rk).

We have the following structure preserving notions:

Homeomorphisim

Bi-Lipschitz

Isometry

8Note that homeomorphism means that two spaces have the same topology,isometry means two spaces have the same geometry.Two measure spaces are measurably isomorphic if there exists a measur-able bijection with a measurable inverse (CSZ: 7.6.17).The Borel sigma algebra of any two polish spaces (complete and separable)are measurably isomorphic iff the spaces have the same cardinality.

A function f : (X, dX) (Y, dY ) is K-Lipschitz if K > 0 s.t.

dY (f(x1), f(x2)) KdX(x1, x2)

and the smallest K > 0 is called the Lipschitz constant.If K = 1 it is a short map, it K (0, 1) it is a contraction (create diagram).Every Lipschitz function is Holder-Continuous of order = 1.A function is Holder-Continuous of order if M > 0 s.t.

dY (f(x1), f(x2)) KdX(x1, x2)

(create diagram)

Uniformly Continuous

Lipschitz

Bounded Derivative

Let Diffp(U,Rn) {f : U Rn : f is p-times differentiable}.C0 Diff1 C1 Diff2 Cp Diffp+1 . . .

9C0

C1

...

Smooth C

From (CSZ, p. 346), we have:

-compact

compact

strongly -compact

From (CSZ, p. 279), we have:

point-wise convergence

uniform convergence on compact sets

uniform convergence

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Topology:f : (X, X) (Y, Y ), it is easier for f to be continuous with a finerdomain or a coarser co-domain.If f is continuous and X X then it will remain continuous with the finertopology X .If f is continuous and Y Y then it will remain continuous with thecoarser topology Y .Whenever X = P(X) is the topology on the domain, every function on thisdomain will be continuous.Whenever Y = {, Y } is the topology on the co-domain, every functioninto this co-domain will be continuous.For a function f : (X, {, X}) (Y,P(Y )) the only continuous functionsare the constant functions.

Given a non-empty set X, let TOP(X) be the collection of all topologies on

X. Since TOP(X) 22X , we have that whenever X is finite TOP(X) willbe finite as well.

{, X}coarsest topology

2Xfinest topology

Using the -order, the indiscrete topology {, X} will always be the -leastset in (TOP(X),), and the discrete topology 2X will be the -greatest set.G.O. (I believe (TOP(X),) is a complete lattice because the intersectionof topologies is a topology.)In the indiscrete topology {, X}: every sequence converges, every set isconnected, every set is compact.In the discrete topology 2X only eventually constant sequences converge, itis difficult for a set to be compact or connected, only finite sets are compact.

Given a non-empty set X consider two topologies (X, 1) and (X, 2) suchthat 1 2. Any sequence that converges in 2 must converge in 1, anyset that is compact in 2 must be compact in 1, any set that is connectedin 2 must be connected in 1.

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ConvexityLet V be a real vector space and let S be any non-empty subset, then:

Aff(S)Affine Hull

span(S)Linear Hull

Conv(S)Convex Hull

Cone(S)

Convex Conic Hull

For a canonical exercise let the vector space be R2 and let S = {e1, e2}.We have: span(S) = R2.Aff(S) = {(x1, 1 x1) : x1 R}Cone(S) = R2+ (the positive orthant)Conv(S) = 1 (the 1-simplex of probabilities)CREATE this figure in MathematicaNote if our set is a singleton (consists of only one vector) S = {x} then:span(S) is a line, Cone(S) is a ray, and Conv(S) = Aff(S) = {x}.

Note that if S is a vector subspace of V it is equal to the intersection of allvector subspaces containing it. If S is a subset of a vector space V , then thesmallest vector subspace that contains S is span(S) which is equal to theintersection of all vector subspaces of V that contain S. The same holds foraffine subspaces, conic hulls and convex hulls.Datorro Convex book p. 65: Conv(S) Aff(S) = Aff(S) = Aff(S) =Aff Conv(S).A set of vectors in Rn are affinely independent if there is no proper affinesubspace which contains all of them.To visualize affine independence consider the following three affinely inde-pendent vectors in R2, along with the affine hulls containing any two of

them:

(02

),

(20

),

(12

)

12

Aff02 ,

20

Aff12 ,

20

Aff12 ,

02

-4 -2 2 4

-6

-4

-2

2

4

6

Notice that no line contains all three points.An intuitive explanation for how a set of n + 1 vectors can be affinely in-dependent in Rn is if we let any one of the points be the new origin, theremaining n points would be linearly independent in Rn.The affine hull of n+ 1 affinely independent vectors in Rn is Rn.

Optimization recall the following:

Critical points of f

Local extrema

Global Extrema

Suppose we wish to maximize an objective function of two variables u(x1, x2)subject to p1x1 + p2x2 w and x1 0, x2 0. We may write:maxu(x1, x2) s.t. p1x1 + p2x2 w, x1 0, x2 0.The corresponding Lagrangian function is:L = u(x1, x2) + (w p1x1 p2x2) + 1x1 + 2x2The Kuhn Tucker conditions tell us we face the following 8 cases:

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Case 1 21 + 0 02 0 + 03 0 0 +4 + + 05 + 0 +6 0 + +7 + + +8 0 0 0

The following diagram illustrates the points these cases correspond to whenp1, p2, w > 0:

12

3

4

56

8-2 -1 1 2

-1.0

-0.5

0.5

1.0

1.5

2.0

Notice that case 7 is impossible because if x1 = x2 = 0 then the first con-straint cannot possibly bind. The only time case 7 would hold is if theconstraint passed the origin.Note: if we know that the objective function u(x1, x2) is increasing in bothof its arguments, then the only relevant cases are: 1, 4, and 5.This diagram was inspired by Nolan Millers handout: The Kuhn TuckerConditions and you.

The following diagram illustrates the relationship between types of con-cavity:

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Hopefully the following diagram helps make things clearer:

strictly quasi-concave

strictly concave quasi concave

concave

A constant function is a function that is concave but not strictly quasi-concave. Are there other functions with this property?An increasing transformation of a concave function is quasi-concave. (Simi-larly: an increasing transformation of a homogeneous function is homothetic,and an increasing transformation of an spm function is q-spm.)The standard normal distribution is an example of a function that is quasi-concave but not concave:

15

-10 -5 0 5 10

0.1

0.2

0.3

0.4

Quasi-concavity is preserved by monotonic transformations.A Cobb-Douglas technology with DRS is concave (thus q-concave) such as

f(x1, x2) = x1/31 x

1/32 .

A Cobb-Douglas technology with IRS is q-concave (but not concave) such

as f(x1, x2) = x2/31 x

2/32 .

From UPenns math-camp we have the following relationships between dif-ferent notions of concavity (which is the previous diagram including pseudo-concavity, which requires differentiability):

strictly quasi-concave

strictly concave quasi concave

concave pseudo concave

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Measure

Given a non-empty set X the collection of all -algebra on X is a com-plete lattice:

{, X}coarsest algebra

2Xfinest algebra

From K.C. Borders Hitchikers guide p. 133 we have:

ring

algebra ring semi ring

algebra

Also look at Border p. 499 for a table of normed Reisz spaces and theirduals.Equivalently we have the following nested sequence of boxes:

-algebra

algebra

ring

semi-ring

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Let A 2X be a collection of subsets of X. Then if A is both a pi-systemand a -system then it is a -algebra.

system

algebra

algebra pi system

For a collection A 2X let pi(A) be the pi-system generated by A and (A)be the -system generated by A and (A) be the -field generated by (A),and f(A) be the field (algebra) generated by A. We have the following re-lation (based on notes by Matias Cattaneo):

(A)

pi(A) (A)

f(A)

Also discuss semi-algebras and monotone classes.The space L0(,F , P ) contains all measurable functions f : (,F , P ) (R,B).A function is simple if its range is finite. Ms(,F , P ) L0(,F , P ) de-notes the set of simple random variables.We have the following:

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L0(,F , P ) all random variables

L1(,F , P ) all integrable random variables (finite mean)

...

L(,F , P ) bounded random variables

Ms(,F , P ) simple random variables

From Lyapunovs inequality we have that p q implies Lq Lp (CSZ: p.371).If p < q then Lq ( Lp iff (,F , P ) is not composed of a finite set of atoms.L0 is an algebra and a lattice (p. 361). Note: L0 may contain unboundedfunctions and uniform convergence implies pointwise convergence.L1(,F , P ) is a vector space and a lattice but not in general an algebra(CSZ: p.366).

The space Cb(M) is an algebra (a vector space that is closed under mul-tiplication), a lattice (closed under and ) and closed under uniform con-vergence and composition with continuous bounded functions on R.

Theorem 7.1.34 (CSZ) says Ms is a dense subset of the MS L0, , and

X : R is measurable iff there exists a sequence {Xn} of simple measur-able functions s.t. , we have Xn() X().So any measurable function can be approximated by a sequence of simplefunctions which converge pointwise.

Conditional ExpectationE[X|{,}] = E[X], and E[X|(X)] = XLebesgues Decomposition Theorem: = atomic + Abs.Cts. + Sing.Cts.

(CSZ p. 401) the set S L1(,F , P ) is L-dominated, if for some Y 0,

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Y L, |X| Y a.e. for all X S.Theorem 7.5.4 says the integral is -continuous on any L-dominated set.The analogous definition holds for L1-dominated or simply dominated.The Dominated Convergence Theorem says: the integral is -continuous onany L1-dominated set.

Uniformly integrable (u.i.)

L1-dominated

L-dominated

Theorem 7.5.9 says the integral is -continuous on any uniformly integrableset.

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Abstract Algebra:

Magma (Binary Algebraic Structure) closed under operation

Semigroup (Associative)

Monoid (Identity)

Group (Inverse)

Abelian Group (Commutative)

Ring, Field, Vector Space over a Field (Linear Algebra in a Nutshell), Alge-bra (just a VS with vector multiplication)Linear Algebra:

Tensor Product

Kronecker Product

Outer Product

Operations on setsGiven X 6= , intersection, unions and symmetric difference are binary al-gebraic structures. : P(X) P(X) P(X), the intersection operation is commutative, as-sociative, and has an identity X, however it is not invertible.The union operation is commutative, associative, and has an identity ,however it is not invertible.(P(X),) is an Abelian group (like addition), the identity is , and every

21

set is its own inverse AA = .

Irving Kaplansky p.9, (P(X),,) is a commutative associative ring withunit X, in which every element is idempotent.

22

Stanislav Anatolyev p. 87: nice exercise on the Best Linear Predictor